4 research outputs found
On Piercing Numbers of Families Satisfying the Property
The Hadwiger-Debrunner number is the minimal size of a piercing
set that can always be guaranteed for a family of compact convex sets in
that satisfies the property. Hadwiger and Debrunner
showed that for all , and equality is attained for . Almost tight upper bounds for for a
`sufficiently large' were obtained recently using an enhancement of the
celebrated Alon-Kleitman theorem, but no sharp upper bounds for a general
are known.
In [L. Montejano and P. Sober\'{o}n, Piercing numbers for balanced and
unbalanced families, Disc. Comput. Geom., 45(2) (2011), pp. 358-364], Montejano
and Sober\'{o}n defined a refinement of the property: satisfies the
property if among any elements of , at least of the
-tuples intersect. They showed that holds for all
; however, this is far from being
tight.
In this paper we present improved asymptotic upper bounds on
which hold when only a tiny portion of the -tuples intersect. In particular,
we show that for sufficiently large, holds with
. Our bound misses the known
lower bound for the same piercing number by a factor of less than .
Our results use Kalai's Upper Bound Theorem for convex sets, along with the
Hadwiger-Debrunner theorem and the recent improved upper bound on
mentioned above.Comment: 9 page
On transversal and -packing numbers in straight line systems on
A linear system is a pair where is a finite
family of subsets on a ground set , and it satisfies that
for every pair of distinct subsets . As an example of a
linear system are the straight line systems, which family of subsets are
straight line segments on . By and we denote the
size of the minimal transversal and the 2--packing numbers of a linear system
respectively. A natural problem is asking about the relationship of these two
parameters; it is not difficult to prove that there exists a quadratic function
holding . However, for straight line system we believe
that . In this paper we prove that for any linear system with
-packing numbers equal to and , we have that
. Furthermore, we prove that the linear systems that attains the
equality have transversal and -packing numbers equal to , and they are a
special family of linear subsystems of the projective plane of order . Using
this result we confirm that all straight line systems with
satisfies .Comment: 22 pages, 7 figure
Helly's Theorem: New Variations and Applications
This survey presents recent Helly-type geometric theorems published since the
appearance of the last comprehensive survey, more than ten years ago. We
discuss how such theorems continue to be influential in computational geometry
and in optimization.Comment: 40 pages, 1 figure
The geometry and combinatorics of discrete line segment hypergraphs
An -segment hypergraph is a hypergraph whose edges consist of
consecutive integer points on line segments in . In this paper,
we bound the chromatic number and covering number of
hypergraphs in this family, uncovering several interesting geometric properties
in the process. We conjecture that for , the covering number
is at most , where denotes the matching number of .
We prove our conjecture in the case where , and provide improved
(in fact, optimal) bounds on for . We also provide sharp
bounds on the chromatic number in terms of , and use them to prove
two fractional versions of our conjecture